40 CFR Part 61 EPA 402-R-98-008 

National Emission Standards 

for Hazardous Pollutants 

Background Information Document
 

Statistical Procedures for Certifying Phosphogypsum for Entry Into Commerce, As
 

Required by Section 61.207 of 40 CFR Part 61, Subpart R
 

November 1998
 

U.S. Environmental Protection Agency
 

Office of Radiation and Indoor Air
 

Washington, DC 20460
 
PREFACE 

The Environmental Protection Agency (EPA) is promulgating revisions of 40 CFR Part 

61, Subpart R, National Emission Standards for Radon Emissions from Phosphogypsum Stacks. 

This Background Information Document (BID) has been prepared in support of the final 

rulemaking. This BID contains an introduction, definitions of statistical terms, an overview, 

details of the methods for certifying phosphogypsum under procedures 1 and 2, and appendix 

explaining the statistics used for certifying phosphogypsum. Byron Bunger, an economist from 

EPA’s Office of Radiation and Indoor Program developed and prepared this BID. 

Copies of this BID, in whole or in part, are available to all interested persons. For 

additional information, contact Eleanor Thornton-Jones at (202) 564-9773 or write to: 

Director, Radiation Protection Division
 

Office of Radiation and Indoor Air (6602J)
 

U.S. Environmental Protection Agency
 

401 M Street, SW
 

Washington, DC 20460
 

ii 
STATISTICAL PROCEDURES FOR CERTIFYING PHOSPHOGYPSUM 

FOR ENTRY INTO COMMERCE, AS REQUIRED BY SECTION 61.207 

OF 40 CFR PART 61, SUBPART R 

Introduction 

This document describes the methods that must be used for certifying phosphogypsum 

for entry into commerce under Sections 61.204 or 61.206, as required by Section 61.207 of 40 

CFR part 61, subpart R–National Emission Standards for Radon Emissions From 

Phosphogypsum Stacks. 

Section 61.204 allows for the lawful removal of phosphogypsum from a stack for 

agricultural purposes if the average radium-226 concentration at the location in the stack from 

which the material is to be removed does not exceed 10 pCi/g and if the administrative 

requirements of the section are satisfied. The demonstration that the average radium-226 

concentrations does not exceed 10 pCi/g must be undertaken before the phosphogypsum can be 

removed from the stack and annually thereafter as long as the removal continues. Section 61.205 

allows for the lawful removal of limited amounts of phosphogypsum from a stack for research 

and development purposes if the administrative requirements of the section are satisfied. This 

usage is not addressed in this document because the radium-226 concentration need not be 

reported under the provisions of this section. Section 61.206 governs the uses of 

phosphogypsum for any purpose other then those allowed by Sections 61.204 and Section 

61.205. This section makes it unlawful to remove phosphogypsum from a stack for any purpose 

except those specified in Section 61.204 or Section 61/205 without the prior approval by EPA. 

Section 61.206 provides for the Administrator to allow for a specific use of phosphogypsum if 

the administrative requirements of the section are satisfied and if the applicant for the permit 

reports the average radium-226 concentration at the location in the stack from which the material 

is to be removed must be reported annually thereafter, as long as the removal continues. 

This document provides the owner or operator of a phosphogypsum stack with the 

necessary techniques to demonstrate that the radium-226 concentration is below 10 pCi/g if the 

removal of phosphogypsum is pursuant to Section 61.204, and to report the radium-226 

concentration of the stack if the removal of phosphogypsum is for the purposes pursuant to 

Section 61.206. 

Definitions of Statistical Terms Used in this Discussion 

Critical value–The per cent critical value of a probability distribution is that value above, 

or below, which only a per cent of the probability lies. Thus there is a .05 probability that a 

normally distributed variable will have a value above the upper 5% critical value, which is 

calculated by summing the product of 1.64 times the standards deviation of the distribution to the 

mean of the distribution. When testing an hypothesis, is the level of significance, and determines 

the critical value. 
Hypothesis testing–A procedure for the statistical determination of the validity of an 

hypothesis. A test statistic, such as the standard normal variable, is calculated for the purpose of 

discriminating between a null hypothesis and an alternative. 

Level of significance–The probability, α, of rejecting the null hypothesis in test of an 

hypothesis. 

Sampling distribution–The probability distribution assumed by a statistic such as the 

sample mean, calculated from a sample drawn from a population. 

Overview 

The procedure for certifying phosphogypsum from an area of a stack for entry into 

commerce under the provisions of Sections 61.204 or 61.206 requires the collection of samples 

of phosphogypsum and the measurement of their radium-226 content. The samples must be 

collected from a regularly spaced locations across the area of the stack being considered for 

entry into commerce. After the radium-226 concentration in each sample is measured, the mean 

and standard deviation of the collected samples must be calculated. 

If the phosphogypsum is to be removed under the provisions of Section 61.204, a test 

must be performed to demonstrate that the radium-226 concentration is below 10 pCi/g. A set of 

decision criteria, based on the sampling distribution for the sample mean, are used to determine 

if the phosphogypsum is acceptable for entry into commerce under this section. The use of these 

criteria require the determination of the critical value for 5% level of significance in the upper, or 

right hand, tail of the sampling distribution. The critical value is 95th percentile of the sampling 

distribution. 

The decision criteria used for certification under Section 61.204 has three outcomes. If 

the critical value is less than or equal to 10 pCi/g, phosphogypsum from this area of the stack can 

be entered into commerce. If the mean of the collected samples is greater than or equal to 

10 pCi/g, phosphogypsum from this area of the stack cannot be entered into commerce. If the 

sample mean is less than 10 pCi/g and the critical value is greater than 10 pCi/g, the 

phosphogypsum cannot be entered into commerce unless further testing demonstrates that a 

recalculated critical value is less than or equal to 10 pCi/g. The sample size must be increased, 

and the sample mean and standard deviation recalculated. The increase sample size reduces the 

standard deviation of the interval between the man of the sampling distribution and critical 

value. This increases the ability of the decision criteria to distinguish between the mean of the 

sample and the 10 pCi/g concentration limit, which improves the chance that the radium-226 

concentration can be shown to be less then 10 pCi/g. A calculation is performed to determine 

the sample size needed to reduce the recalculated critical value to a value no greater than 10 

pCi/g. 

If a larger sample size is needed to demonstrate that the sample mean is less than 10 

pCi/g, the number of additional samples required increases rapidly as the mean approaches 10 

pCi/g, and can be quite large in cases where the sample mean is only slightly less than 10 pCi/g. 

In such cases the cost of gathering additional samples and measuring their radium-226 
concentrations may become a factor in the decision to continue with the attempt to enter the 

phosphogypsum from this area of the stack into commerce for agricultural purposes. 

Any additional samples must also be taken from regularly spaced locations across the 

area of the phosphogypsum stack being considered for entry into commerce. Once the required 

number of additional samples have been collected, the radium-226 concentrations in each sample 

must be measured. The mean and standard deviation of the radium-226 concentrations for the 

entire set of sample concentrations (including those previously measured) must be recalculated 

and a new sampling distribution established. The critical value for a 5% level of significance in 

the upper tail of the new sampling distribution is established. The decision criteria must then be 

revisited. As before, phosphogypsum from this area of the stack can be entered into commerce 

only if the critical value is less than or equal to 10 pCi/g. 

Although acceptance for entry into commerce is the objective of increasing the sample 

size and establishing the new sampling distribution and critical value, and is the expected 

outcome of the reconsideration, it is possible the recalculated critical value will not be less than 

or equal to 

10 pCi/g. This is because random variation in the new sample concentrations, which can result 

from nonuniformity in the distribution of radium-226 may cause an increased sample mean or 

standard deviation. Either or both of these increases can cause the recalculated critical value to 

be larger than had been projected, causing it to be greater than 10 pCi/g. If this is the case either 

the sample size must be increased once again, and a new sampling distribution and critical value 

determined, or the attempt to certify that area of the stack for entry into commerce must be 

abandoned. 

The details of the method for certifying phosphogypsum for entry into commerce under 

Section 61.204 is described in Procedure 1 below. The statistical methods that underlie 

Procedure 1 are discussed in the Appendix. 

If phosphogypsum is to be removed from the stack under the provisions of Section 

61.206, the mean, standard deviation, sample size, and 95th percentile must be reported. The 

method to be used for their calculation is described in Procedure 2 below. 

Procedure 1: Certification of Radium-226 Concentrations Under Section 61.204 

Section 61.204 provides for the distribution and use of phosphogypsum for agricultural 

purposes. Procedure 1 describes the sampling techniques required to satisfy the provisions of 

Section 61.207, when phosphogypsum is distributed and used under Section 61.204. 

The owner or operator of the phosphogypsum stack must demonstrate that the 95th 

percentile of the sample mean for radium-226 is no greater than 10 pCi/g if the phosphogypsum 

is to be introduced in commerce under the provisions of Section 61.204. Steps 1 through 5 are 

required of all applicants. Depending on the outcome of the decision criteria in step 5, further 

analysis may be required in order to meet the conditions for certification. 
(1)	 A minimum of 30 phosphogypsum samples shall be taken at regularly spaced intervals 

across the surface of the location on the stack from which the phosphogypsum will be 

removed. Let n1 represent the number of samples taken. 

(2)	 Measure the radium-226 concentration of each of the n1 samples in accordance with 

analytical procedures described in 40 CFR part 61, appendix B, Method 114. 

(3)	 The mean, x 1 ,and the standard deviation, s1, of the n1 radium-226 concentrations must be 

calculated: 

n1 

∑ xi 

i=1

x 1 = , n1 

s1 = 

x 

n 

i 

i 

n 

1 

1 

1 

1 

1 

− 

− 

= 

∑ ( x ) 

, 

where x 1 and s1 are expressed in pCi/g.
 

(4) Calculate the 95th percentile for the distribution, x *, using the following equation:
 

 

n1 

 , s1  . x = x 1 + 1.64 

 

where x*, is expressed in pCi/g. 

(5) Based on the values for x1 and x*, on of the following conditions will be met: 

(a)	 If x1 < 10 pCi/g and x* # 10 pCi/g; phosphogypsum can be removed from this 

area of the stack for distribution to commerce pursuant to Section 61.204. Report 

the mean, standard deviation, sample size, and 95th percentile. 

(a)	 If x1 > 10 pCi/g; phosphogypsum cannot be removed from this area of the stack 

for distribution to commerce pursuant to Section 61.204. This ends the 

procedure. Certification for this area of the stack is not possible. 

(b)	 If x1 < 10 pCi/g and x*

 > 10 pCi/g, the decision criteria are inconclusive. Further 

sampling is required to determine phosphogypsum can be removed from this area 
of the stack. The objective of the following procedure is to demonstrate, with 

95% probability, that the phosphogypsum from this area of the stack has a 

radium-226 concentration no greater than 10 pCi/g. The procedure is iterative, 

the sample size may have to be increased more than on time. The following 

procedures must be followed if the attempt to certify the phosphogypsum for 

entry into commerce is to be continued; otherwise the phosphogypsum cannot be 

removed from this area of the stack for distribution to commerce under the 

provisions of Section 61-204. Proceed with steps (6) through (11) below. 

(6) Solve the following equation for the total number of samples required: 

 164s1  2 . n = . 2 

 

10 − x 1 

 

The sample is size n2 shall be rounded upwards to the next whole number. The number of 

additional samples needed is nA = n2 – n1. 

(7)	 Obtain the necessary number of additional samples, nA, which shall also be taken at 

regularly spaced intervals across the surface of the location on the stack from which 

phosphogypsum will be removed. 

(8)	 Measure the radium-226 concentration of each of the nA additional samples in accordance 

with the analytical procedures described in 40 CFR part 61, appendix B, Method 114. 

(9)	 Recalculate the mean and standard deviation of the entire set of n2 radium-226 

concentrations by joining this set of nA concentrations with the n1 concentrations 

previously measured. Use the formulas in step (3), substituting the entire set of n2 

samples in place of the n1 samples called for in step (3), thereby determining the mean, 

x 2 , and standard deviation, s2, for the entire set of n2 concentrations. 

(10)	 Repeat the procedure described in step (4), substituting the recalculated mean, x2 for x1, 

the recalculated standard deviation, s2, for s1, and total sample size, n2, for n1. 

(11)	 Repeat the procedure described in step (5), substituting the recalculated mean x 2 for x 1 . 

The test in step (5) may, again, be inconclusive, resulting in outcome (c). This can result 

from nonuniformity in the distribution for radium-226 in phosphogypsum and from the 

random nature of radiation decay. If this is the case, the sample size must again be 

increased, and steps (6) through (11) repeated; otherwise the attempt to certify 

phosphogypsum from this area of the stack must be abandoned. 

Procedure 2: Sampling Radium-226 Concentrations Under Section 61.206 

Uses of phosphogypsum other than those allowed by Sections 61.204 and 61.205 are 

governed by the requirements of Section 61.206. Procedure 2 describes the sampling techniques 
required to satisfy the provisions of Section 61.207, when phosphogypsum is distributed and 

used under Section 61.206. 

A minimum of 30 phosphogypsum samples shall be taken at regularly spaced intervals across the 

surface of the location on the stack from which the phosphogypsum will be removed. 

Measure the radium-226 concentration of each of the samples in accordance with the 

analytical procedures described in 40 CFR part 61, appendix B, Method 114. 

The mean, x, and the standard deviation, s, of the radium-226 concentrations must be 

calculated: 

n 

∑ xi 

i =1

x = , n 

s = 


x 

n 

i 

i 

n 

− 

− 

= 

∑ ( 2 

1 

1 

x) 

, 

where n is the sample size, and 0 and s are expressed in pCi/g. 

Calculate the 95th percentile for the distribution, 0*

, using the following equation: 

 

n 

 , s  * x = 0 + 1.64 

 

where x* is expressed in pCi/g. 

Report the mean, standard deviation, sample size, and 95th percentile. 

APPENDIX
 

Explanation of the Statistics Used for Certifying
 

Phosphogypsum for Agricultural Purposes Under
 

the Provisions of Section 61.204,
 

40 CFR Part 61, Subpart R
 
A series of fundamental theorems in statistical theory proves that the distribution of the 

mean of a sample drawn from any population will approach the normal distribution as the sample 

size increases without limit. The sample mean approaches that of the population from which the 

sample is drawn, and its variance approaches that of the population divided by the sample size. 

The result is that the expected value of x is :, the population mean, and x can be assumed to 

be normally distributed if estimated with a sufficiently large sample. (The expected value is a 

statistical term for the average value of the variable over all possible values of the variable.) The 

variance of the distribution of x is F2/n, where F2, is the population variance. Also, it can 

shown that the sample variance, s2, more closely approximates the population variance, F2, as the 

sample size increases. 

Although all of these results are stated in terms of improved approximations as the 

sample size increases, a minimum sample size of 30 is usually considered large enough that the 

approximations can be assumed to hold. 

The outcome is that the mean of a sample of 30 or more radium-226 concentrations can 

be assumed to be normally distributed with a mean approximately equal the true radium 

concentration, :, and a variance approximately equal the true variance, F2

, divided by the 

sample size, n. 

Figure 1 shows the normal distribution (or density function) with a mean of zero and a 

standard deviation of one. (The standard deviation is the square root of the variance.) This 

particular density function is known as the standard normal distribution. The areas and ordinates 

of this distribution are found in standard statistical tables. The letter z is a symbol often used for 

the variate of this particular distribution. The value of the mean or any percentile (such as the 

95th percentile, which determines the upper 5% tail) from any normal distribution can be 

transformed into a corresponding value for the standard normal distribution and then transferred 

back to the original distribution. This makes it possible to use the tables for the standard normal 

distribution to determine the percentiles for any normally distributed variate. The normal 

distribution has infinitely long tails on the left and right; and the area under this distribution, 

when integrated from negative infinity to positive infinity is one. Therefore, the probability of 

the occurrence of z within any specified range is determined by integrating the area under the 

normal distribution within this range. The areas under the tails of this distribution are most often 

of interest in testing hypotheses. The right tail, shown by the shaded area in Figure 1, includes al 

values of z from 1.64 to infinity. It has an area of 0.05. The value of z equal 1.64 is the 95th 

percentile. In sampling from this distribution, the probability of any single value of z equaling 

or exceeding 1.64 is 5%. 
A major difficulty in determining whether or not the true radium-226 concentration in 

phosphogypsum from an area of a stack is less than or equal 10 pCi/g is the random variation in 

the measured values. This randomness is caused by the uneven distribution of radium-226 in the 

phosphogypsum and the randomness inherent in the measurement of the radium concentration in 

individual samples. The true concentration can never be exactly determined because of this 

randomness. The accuracy of the estimated concentrations can be improved upon only by 

increasing the number of samples used in the estimation. 

The decision making framework for determining whether or not phosphogypsum from an 

area of a stack can be entered into commerce fro agricultural purposes must be designed in 

recognition of this randomness. The method used for structuring this decision making is based 

on a test of hypothesis. The test is designed to accept a hypothetical outcome, or null hypothesis, 

unless there is persuasive evidence that this outcome is incorrect. Depending on the purpose of 

the test, the null hypothesis may be defined to be the expected or usual outcome; or, as in this 

case, the outcome that is not desired. Therefore, the null hypothesis represents the case where 

phosphogypsum is unacceptable for agricultural purposes, and represents a radium-226 

concentration greater than 10 pCi/g. 
The null hypothesis is rejected only if the evidence is persuasive that the true 

concentration is less than 10 pCi/g. This means that the phosphogypsum is determined to be 

acceptable for agricultural purposes only if the radium-226 concentration can be shown to be less 

than 10 pCi/g by means of the statistical test described in this document. 

To illustrate hypothesis testing, assume an area of a hypothetical phosphogypsum stack 

has a true concentration equal 10 pCi/g. This concentration is used because it is the highest 

concentration permissible for use in agriculture, and therefore, the highest concentration not 

meeting the conditions of the null hypothesis. The distribution of the mean of a sample drawn 

from this area would be expected to be normally distribute with a mean of 10 pCi/g. The 

standard deviation of this distribution could be estimated from a sample drawn from this area 

(when dealing with the distribution of the sample mean, the standard deviation is usually termed 

the standard error). Assume, for this illustration, that the standard error is 1.5 pCi/g. (A sample 

of 30 with a standard deviation equal 8.2 pCi/g would yield a standard error of the mean equal 

1.5 pCi/g.) The standard deviations of radium-226 measurements on five phosphogypsum 

stacks, as reported in the document: A Long-Term Study of Radon and Airborne Particulates at 

Phosphogypsum Stacks in Centra Florida, (EPA 520/5-88-021, Oct. 1988), range from about 

1.65 to 18.3 pCi/g, so a standard deviation equal 8.2 pCi/g is reasonable. The sampling 

distribution is show in Figure 2. This distribution is entirely hypothetical, and is used only to 

illustrate this discussion. The shading in the left hand tail is explained below. 

When testing an area of the stack with an unknown radium concentration, the mean of the 

sample can be compared to this distribution to determine how likely it is that the sample was 

drawn from an area of a phosphogypsum stack with a mean concentration greater than 10 pCi/g. 

The null hypothesis would be rejected only if the sample mean were small enough that it is 

unlikely that it could have been drawn from an area with a rue concentration greater than 10 

pCi/g. 

The probability of erroneously rejecting the null hypothesis expresses the likelihood of 

mistakenly concluding that phosphogypsum can be entered into commerce when, in fact its 

radium-226 concentration is greater than 10 pCi/g. This probability must be established before 

the null hypothesis can be tested, and, when decided upon, determines the critical value. 

The critical value divides the sampling distribution into a region for accepting the null 

hypothesis and a region for rejecting the null. 

There are no rules for determining an acceptable probability or erroneously rejecting the 

null hypothesis. The choice should be made in the context of the consequences of an incorrect 

decision. If there are expected to be substantial consequences from incorrectly rejecting the null 

(i.e., concluding the concentration is less than or equal 10 pCi/g when it is, in fact, greater than 

10 pCi/g), then it is reasonable to establish a small chance of error. If the consequences are not 

expected to be substantial, it is reasonable to establish a larger chance of error. 
Continuing with the illustration, assume a 5% probability is decided upon. This 

establishes a critical value equal 7.5 pCi/g, which is calculated as follows: 

 s
 
 x C = :-z 

 

n 

 ,
 

= 10 - 1.645 (1.5), 

= 7.5, 

where x C is the critical value, 

: = 10 pCi/g, 

z = 1.64, based on a 5% probability of erroneously rejecting the null hypothesis, 

and s/ 

The null hypothesis is accepted if the mean of a sample drawn from an area of a pile 

n = 1.5 pCi/g. 
being considered for entry into commerce for agricultural purposes is greater than 7.5 pCi/g, and 

is rejected if the sample mean is less than or equal 7.5 pCi/g. The region of rejection is the 

shaded area to the left of 7.5 pCi/g in Figure 2. 

Therefore, if a sample drawn from an area of a stack with an unknown concentration has 

a mean less than or equal 7.5 and a standard deviation of 8.2, the null hypothesis would be 

rejected, which means that the phosphogypsum is judged to be acceptable for use for agricultural 

purposes. 

Although this illustration is a straightforward application of a test of an hypothesis, the 

test can be structured somewhat more conveniently with some recalculation. This is done by 

determining a different critical value, x *

 , based on the sampling distribution for sample drawn 

from the area of the stack being tested. The new critical value is calculated using the following 

equation: 

 

n 

 , s  * = x x + z 

 

where x * is the 95th percentile, which determines the upper 5% tail area, 

x is the mean of the measured radium concentration (in pCi/g) measured from the 

n samples, 

s	 is the standard deviation of the measured radium concentration (in pCi/g) 

measured from the n samples, 

and z = 1.64, based on an upper tail area equal 5%. 

The critical value has been redefined, and has changed numerical value. Nevertheless, 

the new decision rule has the same outcome as the previously formulated decision rule. This is 

shown in the following equations: 

 s
 
 if x #10 - z 

 

n 

 ,
 

 s
 
 then x + z 

 

n 


 #10,
 

or x * #10. 
The equivalency of these two approaches can been in Figure 3 where the null hypothesis 

is represented by the distribution on the right. The region below 7.5 pCi/g, under the left tail of 

this distribution, represents the rejection region for the null hypothesis using the first definition 

of the critical value. The distribution on the left illustrates an alternative sampling distribution 

with a mean less than 7.5 pCi/g. The mean of the alternative distribution falls in the region for 

rejecting the null hypothesis. A radium-226 concentration of 10 pCi/g falls in the upper tail of 

the alternative distribution. The alternative critical value, x * , bounds the area in the right hand 

tail of the alternative distribution that represents rejection of the alternative hypothesis (and the 

acceptance of the null hypothesis). The shaded area to the right of x * is this region, and x * 

divides the alternative sampling distribution into acceptance and rejection regions. Based on the 

second method for determining the critical value, the null hypothesis is rejected since 10 pCi/g is 

greater than x *. 

The sampling distributions for the null hypothesis and for the alternative have the same 

standard deviation. Considering how the critical value, x * would move to the right if the mean 

of this alternative sampling distribution were moved to the right confirms that a sampling 

distribution with a mean of 7.5 pCi/g would have a critical value, x * falling at 10pCi/g, as 

proven on the previous page. 
Figure 4 shows only the alternative sampling distribution. The decision rule constructed 

thus far states that a sample found to have an x * less than or equal 10 pCi/g would be judged to 

have a true radium-226 concentration less than 10 pCi/g and a sample with an x greater than 10 

pCi/g would be judged to have a true concentration greater than 10 pCi/g. 

However, this decision rule is not complete because it does not address a third 

possibility; that x is less than 10 pCi/g while x * is greater than 10 pCi/g. Even though x is 

less than 10 pCi/g in this case, it is not sufficiently low that there is assurance that the true 

concentration is below 10 pCi/g. Further sampling to increase the sample size is a way of 

reducing the interval between x and x * . This is because the variance of he distribution of the 

sample mean is inversely related to the sample size; therefore, increasing the sample size reduces 

the variance of the sampling distribution, as discussed on the first page of this appendix. With a 

sufficiently large sample this interval can be reduced enough that x * is equal to or less than 10 

pCi/g. Therefore, increasing the sample size increases the ability of the decision rule to 

distinguish between distributions having different means, increasing the likelihood of either 

accepting or rejecting the null hypothesis. 

Figure 5 demonstrates this; the wider, flatter, distribution could represent the sampling 

distribution for the mean concentration based on a sample size 30, and the distribution with the 

higher, narrower, peak could represent the distribution of x for a four fold increase in the 

sample size to 120, which reduces the standard deviation by a factor of 2. In the case shown 
there, the increased sample size reduces the standard deviation of the sampling distribution 

sufficiently that the new critical value, x *

2, is less than 10 pCi/g. 

This is the basis for the provision in Section 61.207(c), which requires further testing by 

increasing the sample size, as discussed in Procedure 1, steps (6) through (11), on p. 6. 

In one sense, Figure 5 is an over simplification because it shows the two distributions 

with the same mean. This is unlikely to occur in practice because any change in the individual 

members of a sample (here an increase in sample size), would be expected to cause changes in 

both the sample mean, and the standard deviation, due to random variations in the observations 

included in the sample., Therefore, the mean of the new sampling distribution would not be 

expected to be exactly the same as the mean of the original sampling distribution, nor would the 

standard deviation of the new sampling distribution be expected to be exactly one half the 

standard deviation of the original to be exactly one half the standard deviation of the original 

sampling distribution. 

The result is that the increased sample size required in Section 61.207(c), and described 

in Procedure 1, steps (6) through (11), does not guarantee that the recalculated value for x * will 

be below 10 pCi/g. /The recalculated sample mean may be larger than the mean previously 

calculated, or the recalculated sample standard deviation may not be as small as expected. If the 

recalculated x * is not less than or equal 10 pCi/g the sample size can again be increased and the 
decision rule reconsidered. In practice the cost of continuing to take samples and revisiting the 

decision rule may become prohibitive and the attempt to certify this area of the stack for the 

removal of phosphogypsum for agricultural purposes may be abandoned. 

Clearly, the closer the true radium concentration is to 10 pCi/g, the greater the required 

reduction in the interval between x and x *; therefore, the greater the required reduction in the 

variance of the sample mean and the larger the size of the sample required. Again, the cost of 

greatly increasing the sample size may become prohibitive, and the attempt to certify this area of 

the stack for distribution to commerce under the provisions of Section 61.204 may be abandoned 

An area of a phosphogypsum stack is rejected for distribution to commerce under the 

provisions of Section 61.207(b)(3), described in Procedure 1, step (5)(b)) on p. 5, if the mean of 

the sample is equal 10 pCi/g, with no provision for reconsideration using an increased sample 

size. The reason is that the only way x * could be made equal 10 pCi/g if x is found to be 

equal 10 pCi/g is that the sample size must be increased without limit. Since an infinitely large 

sample is not possible, this area of the stack cannot be show to have an x * less then or equal 10 

pCi/g.